Question: Vanessa is 5 times as old as Gabriela. Six years ago, Vanessa was 7 times as old as Gabriela. How old is Gabriela now?
Solution: We can use the given information to write down two equations that describe the ages of Vanessa and Gabriela. Let Vanessa's current age be $v$ and Gabriela's current age be $g$ The information in the first sentence can be expressed in the following equation: $v = 5g$ Six years ago, Vanessa was $v - 6$ years old, and Gabriela was $g - 6$ years old. The information in the second sentence can be expressed in the following equation: $v - 6 = 7(g - 6)$ Now we have two independent equations, and we can solve for our two unknowns. Because we are looking for $g$ , it might be easiest to use our first equation for $v$ and substitute it into our second equation. Our first equation is: $v = 5g$ . Substituting this into our second equation, we get: $5g$ $-$ $6 = 7(g - 6)$ which combines the information about $g$ from both of our original equations. Simplifying the right side of this equation, we get: $5 g - 6 = 7 g - 42$ Solving for $g$ , we get: $2 g = 36.$ $g = 18$.